# Tensor product of dual groups

Let $G,H$ be compact abelian groups, $G^*,H^*$ be their Pontryagin duals, $G^*\otimes H^*$ the tensor product of $G^*,H^*$ and $K=(G^*\otimes H^*)^*$. Does the group $K$ have a special name? What is the interpretation of $K$ in terms of $G$ and $H$?

• Neat question! The most direct construction of $K$ from $G$ and $H$ that I can see is as $K=\operatorname{Hom}(G^*,H)$, but that still goes through the dual of one of the groups. May 27, 2014 at 17:14
• One characterization of the tensor product of abelian groups should be that it is the unique (up to equivalence, maybe up to unique equivalence?) bifunctor $\otimes : \text{Ab} \times \text{Ab} \to \text{Ab}$ preserving colimits in both variables and equipped with a coherent isomorphism $\mathbb{Z} \otimes \mathbb{Z} \cong \mathbb{Z}$ (e.g. this isomorphism should play nice with associativity), if I'm not mistaken. So this operation on compact abelian groups should have the dual characterization: it preserves limits in both variables and (coherently) has $S^1$ as a unit. May 27, 2014 at 21:17
• Preserving limits in both variables is a bizarre condition (e.g. it isn't implied by being part of a closed monoidal structure) so I'm not sure there's anything nice to say about this operation. May 27, 2014 at 21:18